There is a book called "Wildcat Cartridges" available though Krause Publishing that can answer a lot of questions about different cartridges. Ackley became well-known for "improving" cartridges (expanding case capacity by straightening case taper and steeper case shoulder angle).
Keep in mind that it may also be a standard factory round. "6mm" before 1964 typically meant 6mm Lee Navy, since there were very few other 6mm cartridges to be found anywhere, much like "7mm" used to refer to 7X57 but now is typically used to refer to 7mm Remington Magnum. There is a 6mm Ackley based on an improved 6mm Lee Navy, but PO Ackley was prone to attaching very similar names to very different cartridges (22 Ackley Magnum, 228 Ackley Magnum; 25 Ackley, 25 Ackley Krag; etc).
The 236 Remington brass is a dead giveaway. There are other wildcats around using 220 Swift brass necked up to 6mm. Since they use the Swift semi-rimmed case, they are strictly speaking not 6mm Lee Navy Improved.
Like many gunsmiths, PO Ackley did many rebarreling jobs, and many customers did not want wildcat chambers. PO Ackley always stamped his name right by the chambering stamp, and a lot of people over the years have assumed that because his name was on a rifle barrel, that it must be chambered for an improved cartridge, when in fact the rifle may have been chambered for just a standard factory chambering.
My problem is this: I know nothing of the Ackley
PO Ackley's books can likely be found at your local library. They are very interesting reading for gun fanatics, probably pretty boring for non-gun lovers. In it you will find descriptions of many cartridges Ackley modified and designed. You will also find a lot of information about gunsmithing and a lot of mythology that has been passed down over the years.